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Theorem fvreseq 6582
Description: Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.) (Proof shortened by AV, 4-Mar-2019.)
Assertion
Ref Expression
fvreseq (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐵𝐴) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem fvreseq
StepHypRef Expression
1 fvreseq0 6580 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐵𝐴𝐵𝐴)) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
21anabsan2 664 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐵𝐴) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wral 3090  wss 3792  cres 5357   Fn wfn 6130  cfv 6135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143
This theorem is referenced by:  wfr3g  7695  tfr3  7778  fseqenlem1  9180  symgfixf1  18240  dchrresb  25436  bnj1536  31523  bnj1253  31684  bnj1280  31687  rdgprc  32288  frr3g  32368  fourierdlem73  41323
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